Zero Dynamics and Stabilization for Linear DAEs

  • Thomas Berger
Conference paper
Part of the Differential-Algebraic Equations Forum book series (DAEF)


We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the asymptotic stability of the zero dynamics and stabilizability. To this end, the concepts of autonomous zero dynamics, transmission zeros, right-invertibility, stabilizability in the behavioral sense and detectability in the behavioral sense are introduced and algebraic characterizations are derived. It is then proved, for the class of right-invertible systems with autonomous zero dynamics, that asymptotic stability of the zero dynamics is equivalent to three conditions: stabilizability in the behavioral sense, detectability in the behavioral sense, and the condition that all transmission zeros of the system are in the open left complex half-plane. Furthermore, for the same class, it is shown that we can achieve, by a compatible control in the behavioral sense, that the Lyapunov exponent of the interconnected system equals the Lyapunov exponent of the zero dynamics.


Differential-algebraic equations Zero dynamics Transmission zeros Right-invertibility Stabilizability Detectability Lyapunov exponent 



I am indebted to Achim Ilchmann (Ilmenau University of Technology) for several constructive discussions.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität HamburgHamburgGermany

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